THE STUDY OF SOME NONLINEAR DYNAMICAL SYSTEMS WITH DISTRIBUTED PARAMETERS

Abstract

In this paper, the oscillations of some nonlinear dynamical systems with specially distributed parameters are studied. The nonlinear part of the equations depends on the speed under polynomial form. Such situations generalize the Van der Pol equations with one, two, three or four parameters. It is the case of the electrical transistorized circuits (Lienard), the Rayleigh equations for nonlinear vibrations in elasticity, the aero-dynamical flutter. Here, besides the elastic force and the harmonic perturbations, we have dumping forces which contain polynomial terms in speed and in this case, the energetic perturbations can generate self-oscillations. For these systems, we study the stability of solutions in the autonomous or non autonomous case, the existence of bifurcations and limit cycles by using the criteria of Hopf and Bendixon and the Liapunov function. By applying the average method or asymptotic developments with respect to the small parameter, the resonance and the self-oscillations for these nonlinear dynamical systems are studied and the trajectories are specified.